1 Camera calibration With OpenCV {#tutorial_camera_calibration}
2 ==============================
4 Cameras have been around for a long-long time. However, with the introduction of the cheap *pinhole*
5 cameras in the late 20th century, they became a common occurrence in our everyday life.
6 Unfortunately, this cheapness comes with its price: significant distortion. Luckily, these are
7 constants and with a calibration and some remapping we can correct this. Furthermore, with
8 calibration you may also determine the relation between the camera's natural units (pixels) and the
9 real world units (for example millimeters).
14 For the distortion OpenCV takes into account the radial and tangential factors. For the radial
15 factor one uses the following formula:
17 \f[x_{distorted} = x( 1 + k_1 r^2 + k_2 r^4 + k_3 r^6) \\
18 y_{distorted} = y( 1 + k_1 r^2 + k_2 r^4 + k_3 r^6)\f]
20 So for an undistorted pixel point at \f$(x,y)\f$ coordinates, its position on the distorted image
21 will be \f$(x_{distorted} y_{distorted})\f$. The presence of the radial distortion manifests in form
22 of the "barrel" or "fish-eye" effect.
24 Tangential distortion occurs because the image taking lenses are not perfectly parallel to the
25 imaging plane. It can be represented via the formulas:
27 \f[x_{distorted} = x + [ 2p_1xy + p_2(r^2+2x^2)] \\
28 y_{distorted} = y + [ p_1(r^2+ 2y^2)+ 2p_2xy]\f]
30 So we have five distortion parameters which in OpenCV are presented as one row matrix with 5
33 \f[distortion\_coefficients=(k_1 \hspace{10pt} k_2 \hspace{10pt} p_1 \hspace{10pt} p_2 \hspace{10pt} k_3)\f]
35 Now for the unit conversion we use the following formula:
37 \f[\left [ \begin{matrix} x \\ y \\ w \end{matrix} \right ] = \left [ \begin{matrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{matrix} \right ] \left [ \begin{matrix} X \\ Y \\ Z \end{matrix} \right ]\f]
39 Here the presence of \f$w\f$ is explained by the use of homography coordinate system (and \f$w=Z\f$). The
40 unknown parameters are \f$f_x\f$ and \f$f_y\f$ (camera focal lengths) and \f$(c_x, c_y)\f$ which are the optical
41 centers expressed in pixels coordinates. If for both axes a common focal length is used with a given
42 \f$a\f$ aspect ratio (usually 1), then \f$f_y=f_x*a\f$ and in the upper formula we will have a single focal
43 length \f$f\f$. The matrix containing these four parameters is referred to as the *camera matrix*. While
44 the distortion coefficients are the same regardless of the camera resolutions used, these should be
45 scaled along with the current resolution from the calibrated resolution.
47 The process of determining these two matrices is the calibration. Calculation of these parameters is
48 done through basic geometrical equations. The equations used depend on the chosen calibrating
49 objects. Currently OpenCV supports three types of objects for calibration:
51 - Classical black-white chessboard
52 - Symmetrical circle pattern
53 - Asymmetrical circle pattern
55 Basically, you need to take snapshots of these patterns with your camera and let OpenCV find them.
56 Each found pattern results in a new equation. To solve the equation you need at least a
57 predetermined number of pattern snapshots to form a well-posed equation system. This number is
58 higher for the chessboard pattern and less for the circle ones. For example, in theory the
59 chessboard pattern requires at least two snapshots. However, in practice we have a good amount of
60 noise present in our input images, so for good results you will probably need at least 10 good
61 snapshots of the input pattern in different positions.
66 The sample application will:
68 - Determine the distortion matrix
69 - Determine the camera matrix
70 - Take input from Camera, Video and Image file list
71 - Read configuration from XML/YAML file
72 - Save the results into XML/YAML file
73 - Calculate re-projection error
78 You may also find the source code in the `samples/cpp/tutorial_code/calib3d/camera_calibration/`
79 folder of the OpenCV source library or [download it from here
80 ](https://github.com/opencv/opencv/tree/3.4/samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp). The program has a
81 single argument: the name of its configuration file. If none is given then it will try to open the
82 one named "default.xml". [Here's a sample configuration file
83 ](https://github.com/opencv/opencv/tree/3.4/samples/cpp/tutorial_code/calib3d/camera_calibration/in_VID5.xml) in XML format. In the
84 configuration file you may choose to use camera as an input, a video file or an image list. If you
85 opt for the last one, you will need to create a configuration file where you enumerate the images to
86 use. Here's [an example of this ](https://github.com/opencv/opencv/tree/3.4/samples/cpp/tutorial_code/calib3d/camera_calibration/VID5.xml).
87 The important part to remember is that the images need to be specified using the absolute path or
88 the relative one from your application's working directory. You may find all this in the samples
89 directory mentioned above.
91 The application starts up with reading the settings from the configuration file. Although, this is
92 an important part of it, it has nothing to do with the subject of this tutorial: *camera
93 calibration*. Therefore, I've chosen not to post the code for that part here. Technical background
94 on how to do this you can find in the @ref tutorial_file_input_output_with_xml_yml tutorial.
99 -# **Read the settings**
100 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp file_read
102 For this I've used simple OpenCV class input operation. After reading the file I've an
103 additional post-processing function that checks validity of the input. Only if all inputs are
104 good then *goodInput* variable will be true.
106 -# **Get next input, if it fails or we have enough of them - calibrate**
108 After this we have a big
109 loop where we do the following operations: get the next image from the image list, camera or
110 video file. If this fails or we have enough images then we run the calibration process. In case
111 of image we step out of the loop and otherwise the remaining frames will be undistorted (if the
112 option is set) via changing from *DETECTION* mode to the *CALIBRATED* one.
113 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp get_input
114 For some cameras we may need to flip the input image. Here we do this too.
116 -# **Find the pattern in the current input**
118 The formation of the equations I mentioned above aims
119 to finding major patterns in the input: in case of the chessboard this are corners of the
120 squares and for the circles, well, the circles themselves. The position of these will form the
121 result which will be written into the *pointBuf* vector.
122 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp find_pattern
123 Depending on the type of the input pattern you use either the @ref cv::findChessboardCorners or
124 the @ref cv::findCirclesGrid function. For both of them you pass the current image and the size
125 of the board and you'll get the positions of the patterns. Furthermore, they return a boolean
126 variable which states if the pattern was found in the input (we only need to take into account
127 those images where this is true!).
129 Then again in case of cameras we only take camera images when an input delay time is passed.
130 This is done in order to allow user moving the chessboard around and getting different images.
131 Similar images result in similar equations, and similar equations at the calibration step will
132 form an ill-posed problem, so the calibration will fail. For square images the positions of the
133 corners are only approximate. We may improve this by calling the @ref cv::cornerSubPix function.
134 It will produce better calibration result. After this we add a valid inputs result to the
135 *imagePoints* vector to collect all of the equations into a single container. Finally, for
136 visualization feedback purposes we will draw the found points on the input image using @ref
137 cv::findChessboardCorners function.
138 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp pattern_found
139 -# **Show state and result to the user, plus command line control of the application**
141 This part shows text output on the image.
142 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp output_text
143 If we ran calibration and got camera's matrix with the distortion coefficients we may want to
144 correct the image using @ref cv::undistort function:
145 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp output_undistorted
146 Then we show the image and wait for an input key and if this is *u* we toggle the distortion removal,
147 if it is *g* we start again the detection process, and finally for the *ESC* key we quit the application:
148 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp await_input
149 -# **Show the distortion removal for the images too**
151 When you work with an image list it is not
152 possible to remove the distortion inside the loop. Therefore, you must do this after the loop.
153 Taking advantage of this now I'll expand the @ref cv::undistort function, which is in fact first
154 calls @ref cv::initUndistortRectifyMap to find transformation matrices and then performs
155 transformation using @ref cv::remap function. Because, after successful calibration map
156 calculation needs to be done only once, by using this expanded form you may speed up your
158 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp show_results
160 The calibration and save
161 ------------------------
163 Because the calibration needs to be done only once per camera, it makes sense to save it after a
164 successful calibration. This way later on you can just load these values into your program. Due to
165 this we first make the calibration, and if it succeeds we save the result into an OpenCV style XML
166 or YAML file, depending on the extension you give in the configuration file.
168 Therefore in the first function we just split up these two processes. Because we want to save many
169 of the calibration variables we'll create these variables here and pass on both of them to the
170 calibration and saving function. Again, I'll not show the saving part as that has little in common
171 with the calibration. Explore the source file in order to find out how and what:
172 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp run_and_save
173 We do the calibration with the help of the @ref cv::calibrateCamera function. It has the following
176 - The object points. This is a vector of *Point3f* vector that for each input image describes how
177 should the pattern look. If we have a planar pattern (like a chessboard) then we can simply set
178 all Z coordinates to zero. This is a collection of the points where these important points are
179 present. Because, we use a single pattern for all the input images we can calculate this just
180 once and multiply it for all the other input views. We calculate the corner points with the
181 *calcBoardCornerPositions* function as:
182 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp board_corners
183 And then multiply it as:
185 vector<vector<Point3f> > objectPoints(1);
186 calcBoardCornerPositions(s.boardSize, s.squareSize, objectPoints[0], s.calibrationPattern);
187 objectPoints.resize(imagePoints.size(),objectPoints[0]);
189 - The image points. This is a vector of *Point2f* vector which for each input image contains
190 coordinates of the important points (corners for chessboard and centers of the circles for the
191 circle pattern). We have already collected this from @ref cv::findChessboardCorners or @ref
192 cv::findCirclesGrid function. We just need to pass it on.
193 - The size of the image acquired from the camera, video file or the images.
194 - The camera matrix. If we used the fixed aspect ratio option we need to set \f$f_x\f$:
195 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp fixed_aspect
196 - The distortion coefficient matrix. Initialize with zero.
198 distCoeffs = Mat::zeros(8, 1, CV_64F);
200 - For all the views the function will calculate rotation and translation vectors which transform
201 the object points (given in the model coordinate space) to the image points (given in the world
202 coordinate space). The 7-th and 8-th parameters are the output vector of matrices containing in
203 the i-th position the rotation and translation vector for the i-th object point to the i-th
205 - The final argument is the flag. You need to specify here options like fix the aspect ratio for
206 the focal length, assume zero tangential distortion or to fix the principal point.
208 double rms = calibrateCamera(objectPoints, imagePoints, imageSize, cameraMatrix,
209 distCoeffs, rvecs, tvecs, s.flag|CV_CALIB_FIX_K4|CV_CALIB_FIX_K5);
211 - The function returns the average re-projection error. This number gives a good estimation of
212 precision of the found parameters. This should be as close to zero as possible. Given the
213 intrinsic, distortion, rotation and translation matrices we may calculate the error for one view
214 by using the @ref cv::projectPoints to first transform the object point to image point. Then we
215 calculate the absolute norm between what we got with our transformation and the corner/circle
216 finding algorithm. To find the average error we calculate the arithmetical mean of the errors
217 calculated for all the calibration images.
218 @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp compute_errors
223 Let there be [this input chessboard pattern ](pattern.png) which has a size of 9 X 6. I've used an
224 AXIS IP camera to create a couple of snapshots of the board and saved it into VID5 directory. I've
225 put this inside the `images/CameraCalibration` folder of my working directory and created the
226 following `VID5.XML` file that describes which images to use:
228 <?xml version="1.0"?>
231 images/CameraCalibration/VID5/xx1.jpg
232 images/CameraCalibration/VID5/xx2.jpg
233 images/CameraCalibration/VID5/xx3.jpg
234 images/CameraCalibration/VID5/xx4.jpg
235 images/CameraCalibration/VID5/xx5.jpg
236 images/CameraCalibration/VID5/xx6.jpg
237 images/CameraCalibration/VID5/xx7.jpg
238 images/CameraCalibration/VID5/xx8.jpg
242 Then passed `images/CameraCalibration/VID5/VID5.XML` as an input in the configuration file. Here's a
243 chessboard pattern found during the runtime of the application:
245 ![](images/fileListImage.jpg)
247 After applying the distortion removal we get:
249 ![](images/fileListImageUnDist.jpg)
251 The same works for [this asymmetrical circle pattern ](acircles_pattern.png) by setting the input
252 width to 4 and height to 11. This time I've used a live camera feed by specifying its ID ("1") for
253 the input. Here's, how a detected pattern should look:
255 ![](images/asymetricalPattern.jpg)
257 In both cases in the specified output XML/YAML file you'll find the camera and distortion
258 coefficients matrices:
260 <camera_matrix type_id="opencv-matrix">
265 6.5746697944293521e+002 0. 3.1950000000000000e+002 0.
266 6.5746697944293521e+002 2.3950000000000000e+002 0. 0. 1.</data></camera_matrix>
267 <distortion_coefficients type_id="opencv-matrix">
272 -4.1802327176423804e-001 5.0715244063187526e-001 0. 0.
273 -5.7843597214487474e-001</data></distortion_coefficients>
275 Add these values as constants to your program, call the @ref cv::initUndistortRectifyMap and the
276 @ref cv::remap function to remove distortion and enjoy distortion free inputs for cheap and low
279 You may observe a runtime instance of this on the [YouTube
280 here](https://www.youtube.com/watch?v=ViPN810E0SU).
282 @youtube{ViPN810E0SU}